Negative Refractive Index in Dielectric Crystals Containing Stoichiometric Rare‐Earth Ions

Double‐negative refractive index materials have attracted sustained experimental and theoretical interest because they can display a range of surprising optical phenomena, including negative Doppler shifts and perfect lensing. Double‐negative indexes have been achieved experimentally in engineered metamaterials; however, these materials become increasingly challenging to fabricate at shorter wavelengths, and at optical wavelengths only 2D negative index materials have been achieved. Here, it is shown that a double‐negative index can occur in a natural material, near narrow optical transitions of dielectric crystals stoichiometric in a rare‐earth ion. Optical measurements of two candidate materials, the magnetically‐ordered erbium crystals, ErCl3 ·6H2O and 7LiErF4, which have ultra‐narrow optical linewidths of 3 GHz and 250 MHz, respectively, in the telecom band are presented. It is shown that the spectral density of 7LiErF4 is sufficient to achieve a negative index at 1530 nm. This material can enable the exploration of negative refractive index effects at optical wavelengths in a truly 3D, natural medium.


Introduction
Double-negative index materials have simultaneously negative permittivity and permeability.The optical properties of these materials are counter-intuitive, such as a reversed Snell's law, with the incident and refracted light on the same side of the surface normal; a reversed Doppler shift; and antiparallel wave and energy propagation. [1] range of applications have been identified for negative refractive index materials, including electromagnetic cloaking, [2] sub-wavelength resonators, novel antennas, and phase compensators. [3]At optical frequencies, a particularly promising application of these materials is perfect lensing: [4] A flat slab of a negative index material can perfectly image a source with resolution limited only by inhomogeneities in the medium, not the diffraction limit.
Since the seminal proposal for a microwave frequency negative index material in 2000, [4,5] negative indices have been achieved in the microwave, terahertz, and optical domains, using metamaterials. [6]or the metamaterial to be treated as homogeneous and achieve low scattering, the meta-atom spacings must be less than the wavelength of the interacting light. [7,8][11][12][13][14] While thin structures can exhibit novel phenomena, [15] effects such as perfect lensing require a three-dimensional material. [4]etamaterials have been the focus of negative refractive index investigations for the past two decades.However, recent work supports the possibility of negative refractive indices in so-called "natural" materials, such as Dirac semi-metals at microwave frequencies [16] and ensembles of ions in solids at optical frequencies, [17] which we consider here.[20][21] The challenge is that strong electric and magnetic responses are required, but the strength of an allowed optical magnetic dipole transition is typically ≈10 −5 weaker than an allowed optical electric dipole transition. [22]This disparity disqualifies most optical transitions for negative indices, as the atomic density needed to achieve a sufficient magnetic response renders the material opaque due to absorption from the electric dipole transition. [18]One exception is the 4f N → 4f N transitions of rare-earth ions in non-centrosymmetric sites in crystalline solids, where these normally parity-forbidden electric dipole transitions are only weakly allowed.This results in comparable electric and magnetic dipole strengths for certain transitions. [23][26] The resulting high spectral density and comparable electric and magnetic resonances make rare-earth ions a promising system for negative indices.
The considerations for achieving negative refractive indices using rare earth ions differ from metamaterials.We enumerate five key differences.1) Crystal lattices with rare-earth ions are inherently microscopic.2) Unavoidable microscopic defects inhomogeneously broaden optical resonances, necessitating highquality single-crystal samples.3) A single optical resonance can have both magnetic and electric dipole moments, so independent magnetic and electric resonant modes need not be overlapped, which can be challenging in metamaterials.4) Rare-earth resonance line strengths and widths are fixed by the crystalline host; the only methods of tuning these properties are to use different hosts, or to change the ion concentration.5) The crystalline host determines the anisotropy of the system, with uniaxial and biaxial hosts both common.An example of a rare-earth ion optical resonance appropriate for negative refraction is shown in Figure 1.
Although rare-earth ions are a good candidate, the requirements of a negative index are still extreme.Theoretical studies predict negative indices in doped rare-earth crystals for high doping concentrations (≳1%) with optical linewidths less than 1 MHz, [20,[27][28][29] an unfeasible linewidth at such concentrations due to strain-broadening of the line by the dopant itself.Other theoretical works propose using electromagnetically induced transparency techniques in a four-level doped rare-earth crystal, but also neglect inhomogeneous broadening. [27,30]A recent proposal suggests instead using a dielectric crystal containing stoichiometric terbium combined with chirality induced by a strong 4f 8 → 4f 7 5d 1 transition. [31]That work predicted a negative index for a feasible inhomogeneous linewidth of 25 MHz, although a suitable material remains to be identified.
Here, we propose that negative refraction is possible in dielectric crystals containing stoichiometric erbium, without the need for strong chirality or optical control fields.We justify erbium as the best rare-earth ion and present initial optical measurements of two materials with good potential to show narrow optical transitions, ErCl 3 •6H 2 O and LiErF 4 .

Theory
Calculating refractive indices from atomic properties requires an effective medium theory to relate the microscopic electric (magnetic) dipole polarizability (magnetizability) to the bulk medium's permittivity (permeability).Due to interactions between individual dipoles and the coupling of electric dipoles to oscillating magnetic fields and vice versa, developing a correct effective medium theory is challenging.Here, we follow the standard simplification [32][33][34] of using an isotropic cubic lattice and assuming the electric and magnetic responses are independent, allowing the Clausius-Mossotti relation to be applied. [34]This simplified model gives sufficient insight into the order-ofmagnitude regime required for negative refraction.
The Clausius-Mossotti relation that relates the electric polarizability ( e ) of a single atomic oscillator and the relative electric permittivity (ϵ r ) is where  is the atomic number density and ϵ 0 is the vacuum permittivity.The electric polarizability can be modeled as a classical damped dipole, where e and m e are the charge and mass of an electron respectively, and  e , f ED ,  e and Δ e are the resonant frequency, oscillator strength, full-width at half-maximum and detuning of the electric dipole resonance, respectively.From Equations ( 1) and ( 2), the real part of the permittivity is minimized at a detuning of and will be negative when Following the same procedure with magnetic analogs of Equations ( 1)-( 4), we obtain a requirement for negative permeability, where f MD ,  m , and  m are the oscillator strength, frequency, and full-width at half-maximum of the magnetic dipole resonance, respectively.The terms in the brackets of Equations ( 4) and ( 5) are proportional to the spectral density of the electric and magnetic dipole resonances, respectively.If an atomic transition is able to simultaneously satisfy both equations, then the medium will have a negative index near resonance.
We now identify which rare-earth ions can satisfy the negative permeability requirement, Equation (5).We do not consider particular crystal hosts yet, as the magnetic dipole oscillator strengths of 4f N → 4f N transitions change little between hosts. [35]We consider transitions from ground states in the near-IR and above; lower energy transition linewidths are strongly non-radiatively broadened.The extreme requirement of Equation (5) limits viable transitions even when bestowing the minimum inhomogeneous linewidth observed in any rare-earth system ( = 10 MHz, Nd:YLiF 4 [25] ) and the maximum theoretical ion density ( = 3.5 × 10 28 m −3 , rare-earth nitride).An exhaustive search of all 4f N → 4f N transitions, made using free-ion magnetic dipole oscillator strengths, [35] identified 22 viable transitions (see Section SI, Supporting Information).This paper focuses on the transition that satisfies Equation ( 5) by the second largest margin, the 4 I 15/2 → 4 I 13/2 transition of erbium.
The 4 I 15/2 → 4 I 13/2 transition of erbium has three properties that further justify its suitability compared to the strongest transition, 5 I 8 → 5 I 7 transition of holmium.First, it has the second narrowest optical linewidth observed in any solid (16 MHz), in Er:YLiF 4 . [26]Second, erbium has a zero-spin nuclear isotope, which is beneficial as hyperfine structure would dilute the spectral density.[38][39][40][41] This is beneficial because when Equations ( 4) and ( 5) are satisfied in a balanced manner, the imaginary component of the refractive index (i.e., the loss) at the bare resonance frequency decreases with increasing spectral density, with loss scaling with the inverse of spectral density when the inequalities are strongly satisfied. [34]aving identified erbium as a strong candidate to satisfy Equation (5), we now consider the achievable spectral density.[44] Satisfying Equation ( 5), therefore, requires  ≲ 1 GHz.
Many erbium systems have shown linewidths below 1 GHz; [26,[45][46][47][48][49][50] however, all these measurements were made in doped systems with erbium concentrations ⩽50 ppm.At higher concentrations, disorder in the lattice from erbium itself broadens the inhomogeneous line by several orders of magnitude. [51]However, crystals with stoichiometric erbium have minimal lattice disorder.In fact, a 25 MHz inhomogeneous linewidth has been observed in the stoichiometric europium crystal EuCl 3 •6H 2 O. [52] Unfortunately, observation of narrow inhomogeneous linewidths in a stoichiometric europium crystal does not guarantee the equivalent in an erbium crystal.Europium has minimal magnetic broadening due to the negligible magnetic moment of its singlet ground state.In contrast, the ground state doublet of erbium has a large magnetic moment, meaning disorder in the electronic spin states broadens the transition via magnetic dipole-dipole interactions.Monte Carlo simulations, detailed in Section SIII, Supporting Information, indicate that magnetic broadening can be suppressed below 1 GHz, provided 99.2% or more of the erbium ions occupy the same spin state.Therefore, it is necessary to cool the crystal below its magnetic ordering temperature (T c ).For crystals whose sole magnetic ion is erbium, T c is typically below 1 K. [53] With the electron spin disorder removed, spectral densities able to satisfy Equations ( 4) and ( 5) are plausible.Further, the Monte Carlo simulations in Section SIII, Supporting Information, show that the spectral densities of doped rare earth systems are fundamentally limited to be three orders of magnitude weaker than stoichiometric crystals, ruling out their potential use in negative index applications.

Experimental Section
It remains to identify stoichiometric erbium crystals that could satisfy the linewidth and balanced electric and magnetic dipole oscillator strengths required for a negative index.The high spectral density of these materials implies very strong absorption, making optical studies of these materials difficult, and no accurate linewidth measurements are available in the literature for any stoichiometric erbium crystal.So, optical transitions between the lowest 4 I 15/2 and 4 I 13/2 levels in two candidate crystals ErCl 3 •6H 2 O and 7 LiErF 4 were measured here.These crystals were chosen because of the narrow optical lines measured in the isostructural crystals Er:YLiF 4 [26] for LiErF 4 and EuCl 3 •6H 2 O [52] for ErCl 3 •6H 2 O.
Due to the high absorption, the transmission and reflection of light incident on one crystal face were both measured.These two measurements are complementary in the high absorption regime: the transmission is zero on resonance, but provided information in the wings of the optical line, whereas the reflection probes the large change in electromagnetic impedance at resonance.The reflection and transmission measurements alone could not determine the permeability and permittivity, which required phase-sensitive measurements, [12] but were sufficient to check the spectral density order-of-magnitude requirements.
An ErCl 3 •6H 2 O crystal was grown a saturated solution of erbium chloride in water, similar to the process described in ref. [52], and cleaved twice along a (100) plane to provide a 2.7 mm thick sample with high-quality surfaces.Reflection and transmission measurements were performed with the crystal submerged in liquid helium at 2 K, above the magnetic ordering temperature, with a tunable laser incident 3°from normal to the cleaved plane.Measurements were taken for light polarized parallel and perpendicular to the crystal C 2 ([010]) axis.The laser power was 400 μW; higher power could not be used as the vibrational modes of H 2 O provide an efficient non-radiative decay mechanism for 4 I 13/2 excitations, [56] which heated the sample and populated higher 4 I 15/2 crystal field levels despite the crystal being immersed in liquid helium.
ErCl 3 •6H 2 O measurements are shown in Figure 2.For ≈10 GHz near resonance, no light was transmitted through the crystal and the reflected signal varied significantly.Outside the opaque region, a beat in the reflected signal occurred due to interference of reflections from the front and back surfaces of the crystal.The beat was suppressed for ⃗ E ⟂ C 2 polarization due to absorption from a strongly polarized broad H 2 O vibrational transition.
The complete absorption near resonance inhibited measurement of the inhomogeneous linewidth using transmission; however, the width of the reflection's dispersive curve indicated a linewidth  ≈ 3 GHz.This was inadequate for Equation ( 5), but was measured at 2 K>T c .Simple Monte-Carlo simulations of magnetic disorder predicted ≈1 GHz of broadening on the like→like spin transitions, indicating that magnetic disorder was the dominant source of line broadening.The disorder could not be removed by cooling ErCl 3 •6H 2 O below the 353 mK critical temperature as the crystal is efflorescent at room temperature  4 I 15/2 level to the lowest 4 I 13/2 level.The Reflection signal has been normalized such that the mean signal in the opaque region equals the reflectance expected from the baseline refractive index, [59] and the transmission signal has been normalized to a maximum of 1. in vacuum, making it unsuitable for mounting in most ultralow temperature refrigeration systems.Nonetheless, there was a large modification to the electromagnetic impedance about resonance, as demonstrated by the reflection intensity changing by 37% and 9% for the two polarizations.A Kramers-Kronig transformation of the ⃗ E ⟂ C 2 reflection was performed to obtain the refractive index; this assumed, as justified in the next paragraph, the change in reflection is purely due to an electric response, [57] which was not valid for ⃗ E ∥ C 2 .The wings of the reflection were reconstructed to remove back crystal surface reflections, [58] giving a change in the refractive index of 0.23, as shown in the Figure 2b inset.
The reflection from the front surface followed the usual dispersion curve for ⃗ E ⟂ C 2 polarized light, [58] yet was reversed for ⃗ E ∥ C 2 polarization.Modeling the reflected signal as the sum of reflections from a non-resonant dielectric medium and an array of resonant dipoles, these curves indicated that ⃗ E ⟂ C 2 was a dominantly electric dipolar interaction, whereas ⃗ E ∥ C 2 was dominantly magnetic dipolar (see Section SII, Supporting Information).While in principle the polarized magnetic dipole oscillator strengths could be calculated to verify this polarization dependence, the oscillator strengths varied considerably about zero field, requiring precise knowledge of the disordered magnetic environment (see Section SIV, Supporting Information).
ErCl 3 •6H 2 O is, therefore, a promising candidate for negative refraction if magnetically ordered.It had comparable electric and magnetic dipole oscillator strengths, and a spectral density that was only a factor of ≈20 below our negative permeability criterion, despite being magnetically disordered.Magnetically ordering this crystal required making a sealed sample holder to prevent efflorescence of the sample, allowing it to be installed in a dilution refrigerator.Typically, such sample holders are made by indium-sealing windows into a small cell filled with an atmosphere of helium, either pre-loaded or supplied by a tube connected to a helium supply outside the cryostat.Further improvements in the linewidth are likely if the methods used to narrow the linewidth in EuCl 3 .6H 2 O are used, such as isotopic purification of chlorine, which reduced the linewidth from 100 to 25MHz. [52]

LiErF 4
LiErF 4 is a tetragonal crystal where erbium occupies a site with S 4 point-symmetry (axis along [001]) in a scheelite configuration.It is antiferromagnetic below T c = 375 mK, [60] and has an erbium concentration of  = 1.4 × 10 28 m −3 and a magnetic dipole oscillator strength of f MD = 1.7 × 10 −7 (see Section SIV, Supporting Information), which requires  < 490 MHz to satisfy Equation ( 5).A sample isotopically purified in 7 Li and with natural erbium isotope abundance grown by the Czochralski method (AC Materials Inc.) was polished close to a (010) plane, 300 μm thick.
Transmission and reflection measurements at normal incidence to the polished surface were performed in a dilution refrigerator with a mixing chamber temperature of 25 mK, below T c .A biased InGaAs photodiode on the coldfinger was used to detect transmitted light, with an optical chopper and lock-in amplifier used to increase sensitivity.Because the cooling power of a dilution refrigerator is much lower compared to immersing the sample in a bath cryostat, a low laser power of 30 nW was required to avoid laser-induced sample heating, despite the lowest 4 I 13/2 level of LiErF 4 having a long lifetime of 2.7 ms with a significant radiative decay channel. [56]The lower laser power resulted in a noisier reflection measurement compared to ErCl 3 •6H 2 O. Measurements were taken in zero applied magnetic field and with 1 T applied along [010].The polarization of the light incident on the sample was unknown, but changes to the polarization using waveplates outside the refrigerator had little effect on the spectra.Results are shown in Figure 3.
The spectra of 7 LiErF 4 are more complex than ErCl 3 •6H 2 O, because its narrower linewidth reveals more structure.When magnetically ordered, an internal magnetic field at the erbium site splits the ground and excited state doublets by 31 and 28 GHz, respectively, with only the lower Zeeman branch of the ground state populated.Here, we focus on the spin like→like transition (see Figure 1), which was less sensitive to magnetic fields and therefore narrower.This transition is opaque across several gigahertz, with  = 250 MHz wide satellite features on the low energy side caused by erbium ions near yttrium substitutional impurities.As the satellite features are perturbed <10 GHz from the main line, the environment of the satellite erbium must be similar to the bulk erbium.Therefore, the main like→like line at 195 932GHz likely has a similar linewidth.
In contrast to the dispersive profile in ErCl 3 •6H 2 O, the reflection near the main like→like line in 7 LiErF 4 dips.Departure from the dispersive profile indicates a large resonant contribution to the bulk refractive index (see Section SII, Supporting Informa- tion), as expected for the narrower linewidth.At zero field the feature width was 1 GHz, while in a 1 T external field the profile changed and had a 240 MHz width (see inset in Figure 3).This matched the linewidth observed in the satellite lines.
The observed 250 MHz inhomogeneous linewidth in 7 LiErF 4 satisfies Equation ( 5), the criterion for negative permeability.Additionally, Gerasimov et al. [40] found the electric dipole transitions contributed most of the absorption intensity over all 4 I 15/2 → 4 I 13/2 transitions in erbium-doped LiYF 4 , suggesting that the negative permittivity criterion Equation ( 4) is also satisfied.However, strong absorption made studying the predicted negative index region unfeasible in the current sample.
Instead, optical properties were estimated using a simple model assuming equal electric and magnetic dipole oscillator strengths, f ED = f MD .This predicts a negative index region with 420 MHz bandwidth and a refractive index, when the real part was minimized, of n = −0.97+ i1.97.This corresponds to an absorption coefficient of  =160 000 cm −1 .At the high energy boundary to the negative index region, the absorption coefficient was  =22000 cm −1 , or 10% transmission through a 1 μm sample.To measure transmission through the negative index region evidently requires either: 1) a  (μm) thick sample, which is experimentally challenging; 2) a higher spectral density, which might be achieved with narrower optical linewidth as detailed below; or 3) a loss-compensating gain mechanism, which could be achieved with additional light fields that optically pump ions to a 4 I 13/2 excited state, similar to the work in refs.[61-63].
Experimentally confirming the negative index would be easier for samples containing a single erbium isotope.In LiYF 4 doped with 2.6 ppm erbium, broadening from natural isotope variation produced a ≈300 MHz linewidth, [64] matching the observed linewidth here.The linewidth of a fully isotopically purified crystal, like 7 Li 168 ErF 4 , should be limited by ≈340 μT of magnetic broadening from the disordered fluorine nuclear moments, [64] which corresponds to less than 5 MHz broadening on the like→like transition.Thus, isotopically purified 7 Li 168 ErF 4 should achieve a spectral density 50 times higher, reducing the absorption at the bare resonance frequency to a level where 50% transmission would occur with a 100 μm sample, with a real index of −2.Er isotopes are readily available with purity above 90%, which is certainly sufficient for this purpose, so growing a suitable isotopically purified crystal is a straightforward exercise.Further improvements of the spectral density might be achieved by careful preparation and mounting of the sample; magnetic inhomogeneity due to the crystal shape will contribute to the inhomogeneous linewidth, and inhomogeneous broadening due to strain on a mounted crystal is a common broadening source in ultra-narrow rare earth crystals.

Discussion
We have shown that LiErF 4 meets the conditions for a negative refractive index, and that ErCl 3 •6H 2 O may meet these conditions when magnetically ordered.These two erbium crystal systems were chosen because ultra-narrow linewidths have been measured in isostructural crystals. [25,26,52]This work only scratches the surface of possible negative refractive index candidate materials.Other crystal hosts would likely be suitable; however, because the limited studies of stoichiometric crystals have primarily focused on energy level structure, [26] little is known about the linewidths attainable and their required crystal growth conditions.As such, finding other suitable materials will rely on empirical measurements of seldom-grown crystals.
Throughout the paper, we have assumed the simple model of Equations ( 4) and ( 5) is correct.In a real atomic medium, this may not hold.The main complications are the interplay between electric and magnetic dipoles belonging to the same resonance, the atomic structure that extends beyond two levels, and multipole interactions.We have also ignored the contribution of other atomic species in the lattice, which will contribute a positive real permittivity, but also enhance the oscillator strength. [65]Thus, our criteria can only be considered as order-of-magnitude requirements.Nonetheless, high-quality stoichiometric rare-earth crystals appear to be excellent candidates for negative indices in an atomic material.Further, stoichiometric crystals present an interesting system for more general fundamental experimental studies of the limits of current models of light-matter interactions, precisely because the simplified models appropriate to weakly interacting systems do not apply here.Detailed experimental characterization of the optical transitions and ion-ion interactions will provide direction for the development of more accurate theoretical models of this interesting regime.
Using stoichiometric rare earth crystals is such a remarkably simple approach to achieving negative refraction it may seem surprising that it has not previously been proposed.We attribute this oversight to the paucity of experimental studies of these concentrated salts, particularly for the high-resolution spectroscopic techniques that can probe close to the atomic resonance, where the extreme properties of these optical transitions are seen.These materials could be attractive candidates for other applications requiring strong light-matter interactions, such as quantum transduction. [66]

Conclusion
We have shown that a natural atomic material, stoichiometric erbium crystals, may achieve simultaneously negative permeability and permittivity at low temperatures.Erbium crystals are particularly well suited for negative indices due to their comparable electric and magnetic dipole moments and narrow optical linewidths.To experimentally demonstrate a negative index, magnetically ordered and likely isotopically pure crystals are needed.Our approach does not require any additional microfabrication or optical infrastructure, unlike metamaterials and other proposed atomic systems.

Figure 1 .
Figure 1.Example of a rare earth optical resonance suitable for negative refraction, on the 4 I 15/2 → 4 I 13/2 transition of Er 3+ .These two states are separated by ≈200 THz due to the spin-orbit coupling of 4f electrons and further split by the electrostatic field of the crystal and the (internal) magnetic field at the erbium site as shown.The resulting like→like transition indicated is a good candidate for a negative refractive index.

Figure 2 .
Figure 2. Transmission and reflection of a 400 μW laser incident on a 2.7 mm thick ErCl3•6H2 O crystal about the transition from the lowest 4 I 15/2 level to the lowest 4 I 13/2 level.The Reflection signal has been normalized such that the mean signal in the opaque region equals the reflectance expected from the baseline refractive index,[59] and the transmission signal has been normalized to a maximum of 1.(a) is measured with incident ⃗ E ∥ C 2 and ⃗ k ⟂ C 2 .(b) is measured with ⃗ E ⟂ C 2 and ⃗ k ⟂ C 2 .(a) inset shows the measurement setup, (b) inset shows the inferred refractive index.
Figure 2. Transmission and reflection of a 400 μW laser incident on a 2.7 mm thick ErCl3•6H2 O crystal about the transition from the lowest 4 I 15/2 level to the lowest 4 I 13/2 level.The Reflection signal has been normalized such that the mean signal in the opaque region equals the reflectance expected from the baseline refractive index,[59] and the transmission signal has been normalized to a maximum of 1.(a) is measured with incident ⃗ E ∥ C 2 and ⃗ k ⟂ C 2 .(b) is measured with ⃗ E ⟂ C 2 and ⃗ k ⟂ C 2 .(a) inset shows the measurement setup, (b) inset shows the inferred refractive index.

Figure 3 .
Figure 3. Transmission and reflection from a 300 μm thick 7 LiErF 4 crystal from the lowest 4 I 15/2 level to the lowest 4 I 13/2 level when anchored to a 25 mK cold finger with no applied magnetic field.The right inset shows the measurement setup; a 30 nW laser of unknown polarization was used with a chopper and lock-in amplifier.The left inset shows the reflection with a 1 T field applied.The magnetic ordering Zeeman-splits the doublets, with only the ground state populated.Color shading indicates absorption due to like→like or like→dislike transitions, as indicated in Figure 1.